Large deformation dynamic analysis of saturated porous media with applications to penetration problems

This paper outlines the development as well as implementation of a numerical procedure for coupled finite element analysis of dynamic problems in geomechanics, particularly those involving large deformations and soil-structure interaction. The procedure is based on Biot’s theory for the dynamic behaviour of saturated porous media. The nonlinear behaviour of the solid phase of the soil is represented by either the Mohr Coulomb or Modified Cam Clay material model. The interface between soil and structure is modelled by the so-called node-to-segment contact method. The contact algorithm uses a penalty approach to enforce constraints and to prevent rigid body interpenetration. Moreover, the contact algorithm utilises a smooth discretisation of the contact surfaces to decrease numerical oscillations. An Arbitrary Lagrangian–Eulerian (ALE) scheme preserves the quality and topology of the finite element mesh throughout the numerical simulation. The generalised-α method is used to integrate the governing equations of motion in the time domain. Some aspects of the numerical procedure are validated by solving two benchmark problems. Subsequently, dynamic soil behaviour including the development of excess pore-water pressure due to the fast installation of a single pile and the penetration of a free falling torpedo anchor are studied. The numerical results indicate the robustness and applicability of the proposed method. Typical distributions of the predicted excess pore-water pressures generated due to the dynamic penetration of an object into a saturated soil are presented, revealing higher magnitudes of pore pressure at the face of the penetrometer and lower values along the shaft. A smooth discretisation of the contact interface between soil and structure is found to be a crucial factor to avoid severe oscillations in the predicted dynamic response of the soil.     

Finite element simulation of the thermoviscoplastic behaviour of bituminous concrete

The general framework of the paper deals with the finite element modelling of thermomechanical problems involving viscous materials. The study focuses on the statement of constitutive equations describing the thermoviscoplastic behaviour of bituminous concrete, as well as on their implementation in a finite element program. After stating the general equations of the space- and time-continuous problem and the constitutive relations governing the viscoplastic component of the bituminous concrete behaviour, we deal with their integration over finite time steps, considering two different schemes. Eventually, two sets of numerical results are presented. The first one, an homogeneous triaxial test, is used to compare those schemes, whereas the second one consists of numerical simulations of real-size experiments performed on a road structure subjected to thermal and mechanical loadings. By comparing the numerical results with experimental ones, it allows us to test the finite element code on a more complex and realistic problem. Copyright © 1999 John Wiley & Sons Ltd.     

Explicit stress integration with error control for the Barcelona Basic Model: Part I: Algorithms formulations

The numerical integration of the stress–strain relationship is an important part of many finite element code used in geotechnical engineering. The integration of elasto-plastic models for unsaturated soils poses additional challenges associated to the presence of suction as an extra constitutive variable with respect to traditional saturated soil models. In this contribution, a range of explicit stress integration schemes are derived with specific reference to the Barcelona Basic Model (BBM), which is one of the best known elasto-plastic constitutive models for unsaturated soils. These schemes, however, do not address possible non-convexity of the loading collapse (LC) curve and neglect yielding on the suction increase (SI) line. The paper describes eight Runge–Kutta methods of various orders with adaptive substepping as well as a novel integration scheme based on Richardson extrapolation. The algorithms presented also incorporate two alternative error control methods to ensure accuracy of the numerical integration. Extensive validation and comparison of different schemes are presented in a companion paper. Although the algorithms presented were coded for the Barcelona Basic Model, they can be easily adapted to other unsaturated elasto-plastic models formulated in terms of two independent stress variables such as net stress and suction.     

Analysis of one-dimesional wave propagation in a fluid-saturated finite soil column

Biot's dynamic equations of motion for one-dimensional wave propagation in a fluid-saturated linear elastic isotropic soil are solved using Laplace transformation followed by numerical inversion and the results compared with a direct finite element formulation. A soil column of finite dimension subjected to velocity boundary conditions is analysed, allowing for reflection of waves from boundaries. Comparison of time histories at given points along the column shows that the finite element solution gives good agreement with the Laplace transform solution for low as well as high drag.     

Finite element formulation and algorithms for unsaturated soils. Part I: Theory

This paper presents a complete finite‐element treatment for unsaturated soil problems. A new formulation of general constitutive equations for unsaturated soils is first presented. In the incremental stress–strain equations, the suction or the pore water pressure is treated as a strain variable instead of a stress variable. The global governing equations are derived in terms of displacement and pore water pressure. The discretized governing equations are then solved using an adaptive time‐stepping scheme which automatically adjusts the time‐step size so that the integration error in the displacements and pore pressures lies close to a specified tolerance. The non‐linearity caused by suction‐dependent plastic yielding, suction‐dependent degree of saturation, and saturation‐dependent permeability is treated in a similar way to the elastoplasticity. An explicit stress integration scheme is used to solve the constitutive stress–strain equations at the Gauss point level. The elastoplastic stiffness matrix in the Euler solution is evaluated using the suction as well as the stresses and hardening parameters at the start of the subincrement, while the elastoplastic matrix in the modified Euler solution is evaluated using the suction at the end of the subincrement. In addition, when applying subincrementation, the same rate is applied to all strain components including the suction. Copyright © 2003 John Wiley & Sons, Ltd.     

A non‐linear coupled finite element–boundary element model for the prediction of vibrations due to vibratory and impact pile driving

This paper presents a non‐linear coupled finite element–boundary element approach for the prediction of free field vibrations due to vibratory and impact pile driving. Both the non‐linear constitutive behavior of the soil in the vicinity of the pile and the dynamic interaction between the pile and the soil are accounted for. A subdomain approach is used, defining a generalized structure consisting of the pile and a bounded region of soil around the pile, and an unbounded exterior linear soil domain. The soil around the pile may exhibit non‐linear constitutive behavior and is modelled with a time‐domain finite element method. The dynamic stiffness matrix of the exterior unbounded soil domain is calculated using a boundary element formulation in the frequency domain based on a limited number of modes defined on the interface between the generalized structure and the unbounded soil. The soil–structure interaction forces are evaluated as a convolution of the displacement history and the soil flexibility matrices, which are obtained by an inverse Fourier transformation from the frequency to the time domain. This results in a hybrid frequency–time domain formulation of the non‐linear dynamic soil–structure interaction problem, which is solved in the time domain using Newmark's time integration method; the interaction force time history is evaluated using the θ‐scheme in order to obtain stable solutions. The proposed hybrid formulation is validated for linear problems of vibratory and impact pile driving, showing very good agreement with the results obtained with a frequency‐domain solution. Linear predictions, however, overestimate the free field peak particle velocities as observed in reported field experiments during vibratory and impact pile driving at comparable levels of the transferred energy. This is mainly due to energy dissipation related to plastic deformations in the soil around the pile. Ground vibrations due to vibratory and impact pile driving are, therefore, also computed with a non‐linear model where the soil is modelled as an isotropic elastic, perfectly plastic solid, which yields according to the Drucker–Prager failure criterion. This results in lower predicted free field vibrations with respect to linear predictions, which are also in much better agreement with experimental results recorded during vibratory and impact pile driving. Copyright © 2008 John Wiley & Sons, Ltd.     

Modeling piecewise planar fault discontinuities without element-partitioning in 3D reservoir-geomechanical models

Using the extended finite element method, faults can be introduced into a three-dimensional reservoir-geomechanical model without meshing to assess the potential for fault reactivation associated with industrial activities such as disposal of CO₂ or fluid extraction. The residuals to the governing equations include basis functions formed from the enrichment functions for strong and weak discontinuities. The traditional spatial integration scheme is based on an algorithm that partitions each enriched element into a collection of subtetrahedra, which is time and storage intensive. To avoid element partitioning, we adopt the homogeneous numerical integration scheme (referred as HNI hereafter) where the integration of homogeneous monomials over each polyhedron is converted into the integration of the same monomials over the one-dimensional edges of the polyhedron by using Stokes's theorem and Euler's homogeneous function theorem. The integrands for the strong and weak discontinuities are derived to implement the integration scheme in the three-dimensional reservoir-geomechanical model. An example of a jagged fault is presented to reveal the advantages of HNI scheme over the standard element partitioning approach. Several other examples that involve fluid flow, fault sliding, and fault sliding triggered by injection pressure are also presented to demonstrate that accurate and efficient computations are realized by the new integration scheme.     

A coupled two‐phase fluid flow and elastoplastic deformation model for unsaturated soils: theory,implementation, and application

Although numerous numerical models have been proposed for simulating the coupled hydromechanical behaviors in unsaturated soils, few studies satisfactorily reproduced the soil–water–air three‐phase coupling processes. Particularly, the impacts of deformation dependence of water retention curve, bonding stress, and gas flow on the coupled processes were less examined within a coupled soil–water–air model. Based on our newly developed constitutive models (Hu et al., 2013, 2014, 2015) in which the soil–water–air couplings have been appropriately captured, this study develops a computer code named F²Mus3D to investigate the coupled processes with a focus on the above impacts. In the numerical implementation, the generalized‐α time integration scheme was adopted to solve the equations, and a return‐mapping implicit stress integration scheme was used to update the state variables. The numerical model was verified by two well‐designed laboratory tests and was applied for modeling the coupled elastoplastic deformation and two‐phase fluid flow processes in a homogenous soil slope induced by rainfall infiltration. The simulation results demonstrated that the numerical model well reproduces the initiation of a sheared zone at the toe of the slope and its propagation toward the crest as the rain infiltration proceeds, which manifests a typical mechanism for rainfall‐induced shallow landslides. The simulated plastic strain and deformation would be remarkably underestimated when the bonding stress and/or the deformation‐dependent nature of hydraulic properties are ignored in the coupled model. But on the contrary, the negligence of gas flow in the slope soil results in an overestimation of the rainfall‐induced deformation. Copyright © 2015 John Wiley & Sons, Ltd.     

A time‐discontinuous Galerkin method for the dynamical analysis of porous media

We present a time‐discontinuous Galerkin method (DGT) for the dynamic analysis of fully saturated porous media. The numerical method consists of a finite element discretization in space and time. The discrete basis functions are continuous in space and discontinuous in time. The continuity across the time interval is weakly enforced by a flux function. Two applications and several numerical investigations confirm the quality of the proposed space–time finite element scheme. Copyright © 2006 John Wiley & Sons, Ltd.     

Analysis of stone-column reinforced foundations

A numerical model is proposed to analyse elastic as well as elastoplastic behaviour of stone-column reinforced foundations. The stone-columns are assumed to be dispersed within the in situ soil and a homogenization technique is invoked to establish equivalent material properties for in situ soil and stone-column composite. The difficulties encountered in carrying out elastoplastic analyses of composite materials are overcome by adopting a separate yield function for each of the constituent materials and a sub-iteration procedure within an implicit backward Euler stress integration scheme. In the proposed procedure, equilibrium as well as kinematic conditions implied in the homogenization procedure are satisfied for both elastic as well as elastoplastic stress states. The proposed model is implemented in an axi-symmetric finite element code and numerical prediction is made for the behaviour of model circular footings resting on stone-column reinforced foundations. This prediction indicates good agreement with experimental observation. Finally, a new scheme in which the length of stone-column is variable is proposed and its behaviour is examined through a numerical example. © 1998 John Wiley & Sons, Ltd.     

A coupled finite element–rigid block method for transient analysis of rock caverns

A coupled finite element–rigid block model for the transient analysis of caverns in jointed media is presented. This coupling permits the modelling of lined openings in a jointed rock mass as well as the propagation of stress waves to the cavern. Both the finite element and the rigid block algorithms employ explicit time integration; an efficient, stable scheme is developed for coupling the two algorithms. Two numerical examples are given: one is a simple validation, the second is a representation of a lined cavern in a sparsely jointed medium.     

A NON-LINEAR SEISMIC RESPONSE ANALYSIS METHOD FOR SATURATED SOIL–STRUCTURE SYSTEM WITH ABSORBING BOUNDARY

A non-linear seismic response analysis method for 2-D saturated soil–structure system with an absorbing boundary is presented. According to the 3-D strain space multimechanism model for the cyclic mobility of sandy soil, a constitutive expression for the plane strain condition is first given. Next, based on Biot's two-phase mixture theory, the finite element equations of motion for a saturated soil–structure system with an absorbing boundary during earthquake loadings are derived. A simulation of the shaking table test is performed by applying the proposed constitutive model. The effectiveness of the absorbing boundary is examined for the 2-D non-linear finite element models subjected to random inputs. Finally, a numerical seismic response analysis for a typical saturated soil–structure system is performed as an application of the proposed method.     

Hybrid FE‐DQ for dynamic analysis of multilayered half‐space subjected to concentrated point impulsive loading

A hybrid finite element method and differential quadrature method (DQM) is developed to estimate the dynamic response of two‐dimensional multilayered half‐spaces subjected to impulsive point loading. Nonreflecting absorbing boundary conditions consist of appropriate springs, and dampers are considered. The capabilities of the finite element method for solving boundary value problems with general domain, loading and systematic boundary treatment are combined with accurate and stable time marching capabilities of the DQM to develop an accurate and efficient numerical technique. The capability, efficiency, robustness and convergence of the DQM for solving the dynamic problem are demonstrated through numerical simulations of various half‐spaces with different time increments and layer arrangement. Also, comparison study when using Newmark's time integration scheme for the same problem is done. It can be concluded that the DQM as an unconditionally stable method is suitable for solving such a problem. Also, parametric study is performed to show the effect of the absorbing boundary conditions on the dynamic response. Copyright © 2012 John Wiley & Sons, Ltd.     

Effect of compressible parameters on vertical vibration of an elastic pile in multilayered poroelastic media

This paper mainly investigates the influences of compressible parameters on the vertical vibration of a pile embedded in layered poroelastic soil media. The pile is treated as a 1D elastic bar by the finite element method, and fundamental solutions for the layered poroelastic soils due to a vertical dynamic load are obtained by the analytical layer element method. Based on the compatibility conditions, the pile-soil dynamic interaction problem is solved. The numerical scheme has been compiled into a Fortran program for numerical calculation. Influences of the pile-soil stiffness ratio, compressible parameters, vibration frequency and the soil stratification are discussed.     

An unconditionally stable explicit and precise multiple timescale finite element modeling scheme for the fully coupled hydro-mechanical analysis of saturated poroelastic media

An unconditionally stable, fully explicit and highly precise multiple timescale finite element modeling scheme is described for a fully coupled hydro-mechanical (FCHM) analysis of saturated poroelastic media. The finite element method (FEM) is used for the discretization of the FCHM differential equation in the space domain. Direct integration is performed based on the precise time step integration method (PTSIM) for the time derivatives. Two configurations for the proposed scheme are constructed (abbreviated as PTSIM-f1 and -f2, respectively). The stability and convergence of the PTSIM-f1 and -f2 are proved using a matrix-based spectral analysis in the time domain. It is demonstrated that the explicit scheme proposed in this paper is unconditionally stable and independent of the time-step size. The algorithmic error estimation results indicate that the numerical modeling performed using PTSIM-f1 and -f2 in the time domain match the computer precision. Theoretically, the algorithmic error is caused by only the mesh discretization. Therefore, the proposed modeling scheme is a semi-analytical scheme. The applicability and accuracy of the proposed scheme are examined using sample calculations. By comparing with the analytical solutions, it is indicated that the modeling results have significant advantages over the standard FEM in terms of precision and computational efficiency for large timescales.     

Importance of footing–soil separation on dynamic stiffness of piled embedded footings

Different phenomena such as soil consolidation, erosion, and scour beneath an embedded footing supported on piles may lead to loss of contact between soil and the pile cap underside. The importance of this separation on the dynamic stiffness and damping of the foundation is assessed in this work. To this end, a numerical parametric analysis in the frequency domain is performed using a rigorous three‐dimensional elastodynamic boundary element–finite element coupling scheme. Dimensionless plots relating dynamic stiffness functions computed with and without separation effects are presented for different pile–soil configurations. Vertical, horizontal and rocking modes of oscillation are analyzed for a wide range of dimensionless frequencies. It is shown that the importance of separation is negligible for frequencies below those for which dynamic pile group effects start to become apparent. Redistribution of stiffness contributions between piles and footing is also addressed. Copyright © 2009 John Wiley & Sons, Ltd.     

A boundary integral method for steady flow in unsaturated porous media

The governing equation for steady flow in a homogeneous, partially saturated, porous medium can be written in a linear form if one adopts a hydraulic conductivity function which varies exponentially with capillary-pressure head. The resulting linear field equation is a steady Fokker–Planck equation and is well-suited to numerical solution by the boundary integral equation method (BIEM). The exponential conductivity function is often used in soil physics and is known to be a reasonable approximation over limited ranges of pressure head. A computer code based on the BIEM for obtaining numerical solutions is described and tested. The BIEM is found to exhibit quadratic convergence with element size reduction on smooth solutions and on singular problems, if mesh grading is used. Agreement between results from the BIEM code and a finite element code that solves the fully non-linear problem is excellent, and is achieved at a substantial advantage in computer processing time. As an illustrative example, the code is applied to determine the distribution of moisture in the vicinity of a tunnel.     

An efficient finite–discrete element method for quasi‐static nonlinear soil–structure interaction problems

An efficient finite–discrete element method applicable for the analysis of quasi‐static nonlinear soil–structure interaction problems involving large deformations in three‐dimensional space was presented in this paper. The present method differs from previous approaches in that the use of very fine mesh and small time steps was not needed to stabilize the calculation. The domain involving the large displacement was modeled using discrete elements, whereas the rest of the domain was modeled using finite elements. Forces acting on the discrete and finite elements were related by introducing interface elements at the boundary of the two domains. To improve the stability of the developed method, we used explicit time integration with different damping schemes applied to each domain to relax the system and to reach stability condition. With appropriate damping schemes, a relatively coarse finite element mesh can be used, resulting in significant savings in the computation time. The proposed algorithm was validated using three different benchmark problems, and the numerical results were compared with existing analytical and numerical solutions. The algorithm performance in solving practical soil–structure interaction problems was also investigated by simulating a large‐scale soft ground tunneling problem involving soil loss near an existing lining. Copyright © 2011 John Wiley & Sons, Ltd.     

Mixed unsplit-field perfectly matched layers for transient simulations of scalar waves in heterogeneous domains

We discuss a new formulation for transient scalar wave simulations in heterogeneous semi-infinite domains. To deal with the semi-infinite extent of the physical domains, we introduce truncation boundaries and adopt perfectly matched layers (PMLs) as the boundary wave absorbers. Within this framework, we develop a new mixed displacement-stress (or stress memory) finite element formulation based on unsplit-field PMLs. We use, as typically done, complex-coordinate stretching transformations in the frequency domain, and recover the governing partial differential equations in the time-domain through the inverse Fourier transform. Upon spatial discretization, the resulting equations lead to a mixed semi-discrete form, where both displacements and stresses (or stress histories/memories) are treated as independent unknowns. We propose approximant pairs, which, numerically, are shown to be stable. The resulting mixed finite element scheme is relatively simple and straightforward to implement, when compared against split-field PML techniques. It also bypasses the need for complicated time integration schemes that arise when recent displacement-based formulations are used. We report numerical results for 1D and 2D scalar wave propagation in semi-infinite domains truncated by PMLs. We also conduct parametric studies and report on the effect the various PML parameter choices have on the simulation error.